Why the concentration definition matters
In a target-mediated drug disposition (TMDD) model, drug can circulate as free drug or in a drug–target complex. These forms are related, but they are not interchangeable measurements. A concentration reported as “drug in plasma” may represent only free drug, or it may include drug released from complexes during sample processing. Because binding and complex removal can change over time, those choices can affect the apparent concentration profile and the interpretation of nonlinear pharmacokinetics.
The essential bookkeeping is straightforward: count the drug molecules in each form, use the same compartment and units, and include every form the assay actually detects. The resulting total is a drug-equivalent concentration, not the concentration of all molecular species in the sample.
This definition concerns drug in the specified compartment, such as plasma. It does not by itself give total drug in the body: a body-wide amount requires accounting for the amounts in all relevant compartments and tissues. Also, the addition assumes compatible units and a common sampled volume. Adding a plasma concentration to a tissue concentration, or adding values reported in incompatible units, has no such interpretation.
From model states to total drug
A full TMDD model commonly represents free drug, free target, and complex as separate states. Denote free target concentration by \(C_R\). Under 1:1 binding, association transfers drug from the free state to the complex state, and dissociation transfers it back. These processes change the two states in opposite directions. They do not create or destroy drug molecules merely by moving them between forms.
For a compact derivation, suppose all three concentrations refer to one well-mixed compartment of constant volume. Let \(k_{\mathrm{on}}\) be the association rate constant, \(k_{\mathrm{off}}\) the dissociation rate constant, \(k_{\mathrm{el}}\) the first-order elimination rate constant for free drug, and \(k_{\mathrm{int}}\) the first-order rate at which complex leaves the measured compartment through internalization. Let \(u(t)\) be drug input expressed as concentration per unit time. In this simplified illustration, the drug-state equations are:
Adding these equations gives the total-drug balance. The association and dissociation terms cancel because each binding event moves one drug molecule between the two states:
Thus, binding alone changes the partition between free and bound drug, but not the sum. In this example, total drug changes through input and removal of drug in either free or complex form. Actual models may include additional processes, compartments, or recycling. If internalized drug is returned to the measured compartment, that return must be represented in the relevant balance rather than treated as permanent loss.
Worked Example: Converting Free and Bound Drug to Total
Suppose a hypothetical plasma sample contains \(2.40\ \mathrm{nM}\) free drug and \(1.60\ \mathrm{nM}\) drug in a 1:1 target complex. Calculate total drug concentration in drug-equivalent molar units.
Using the definition, add the free concentration and the concentration of drug molecules in complex:
The sample therefore contains \(4.00\ \mathrm{nM}\) total drug equivalents: \(2.40\ \mathrm{nM}\) free and \(1.60\ \mathrm{nM}\) bound. The target molecules in complex are not another \(1.60\ \mathrm{nM}\) of drug. Counting them again would double-count the drug already represented by \(C_{LR}\).
Concentration definitions and plasma measurements
The model state and the reported observation must refer to the same chemical quantity. A free-drug assay aims to quantify drug not bound to target under the specified sampling and assay conditions. A total-drug assay aims to quantify free drug plus drug that was in complex, often by dissociating complexes during sample processing. A complex assay measures the drug–target complex itself. These are different observation targets, even when they are derived from the same sample.
For a 1:1 complex, a total-drug result can be compared with \(C_L+C_{LR}\), while a complex-specific result can be compared with \(C_{LR}\). A free-selective result can be compared with \(C_L\), provided the assay preserves or reliably reconstructs the free fraction. In practice, sample handling, dilution, incubation, and assay reagents can shift binding equilibria or cause dissociation. The intended measurement should therefore be verified from the assay procedure, not inferred from a label such as “plasma drug.”
| Reported quantity | Model quantity it targets | Interpretation |
|---|---|---|
| Free drug | \(C_L\) | Drug not bound to target in the specified compartment |
| Complex-bound drug | \(C_{LR}\) | Drug molecules present in drug–target complex |
| Total drug equivalents | \(C_L+C_{LR}\) | Free drug plus drug carried in complex, for 1:1 binding |
Worked Example: Converting Total Drug to Mass Units
A hypothetical total-drug assay reports \(5.20\ \mathrm{nM}\) drug equivalents. The compound has a molecular weight of \(150{,}000\ \mathrm{g/mol}\). Convert the result to \(\mathrm{mg/L}\).
First convert nanomoles per liter to moles per liter, then multiply by the molecular weight:
The total-drug result is \(0.780\ \mathrm{mg/L}\) in drug equivalents. This conversion uses the molecular weight of the drug, not the molecular weight of the whole drug–target complex: the assay result counts drug molecules, including those carried in complex.
How binding changes the free-to-total relationship
The relationship between free and total concentration is generally not a constant multiplier. As one illustrative limiting case, suppose binding is at rapid equilibrium, is 1:1, and the total concentration of available target \(C_{R,\mathrm{tot}}\) is fixed. Let \(K_D\) be the equilibrium dissociation constant. The equilibrium complex concentration is
Substituting this expression into the total-drug definition yields:
This relation illustrates why equal changes in free drug need not correspond to equal changes in total drug. When free concentration is small relative to \(K_D\), the target-bound pool can change substantially as free drug rises. As free concentration becomes large relative to \(K_D\), the available target becomes increasingly occupied, and the bound pool approaches its upper limit \(C_{R,\mathrm{tot}}\). This equilibrium expression is an approximation with stated assumptions; it is not a substitute for the dynamic full TMDD model when binding, target turnover, or complex removal cannot be treated as rapid equilibrium.
Worked Example: Nonlinear Mapping From Free to Total
Consider a hypothetical system satisfying the rapid-equilibrium assumptions above, with \(C_{R,\mathrm{tot}}=10.0\ \mathrm{nM}\) and \(K_D=2.00\ \mathrm{nM}\). Calculate total drug at free concentrations of \(1.00\ \mathrm{nM}\) and \(10.0\ \mathrm{nM}\).
At \(C_L=1.00\ \mathrm{nM}\), the complex concentration is:
Therefore, \(C_{L,\mathrm{tot}}=1.00+3.33=4.33\ \mathrm{nM}\). At \(C_L=10.0\ \mathrm{nM}\):
Free drug increased tenfold, from \(1.00\) to \(10.0\ \mathrm{nM}\), while total drug increased about 4.23-fold, from \(4.33\) to \(18.3\ \mathrm{nM}\). The difference arises because the complex-bound pool increased but approached its \(10.0\ \mathrm{nM}\) target-limited maximum. These calculations describe the specified equilibrium illustration, not a claim that a real time-course follows equilibrium at every instant.
Interpreting profiles and checking for mismatches
Free concentration is the model state for unbound drug, whereas total concentration includes the bound pool. Consequently, a total-drug profile can rise or fall differently from the free-drug profile when complex formation, dissociation, or removal changes over time. Total drug may remain relatively high while free drug falls if drug remains in a persistent complex pool. Conversely, rapid complex removal can reduce total drug even when free drug changes little over a short interval. These patterns do not, on their own, identify which process caused the difference.
Before interpreting a concentration–time profile, check that the observation definition matches the model prediction. Useful checks include:
- Confirm whether the assay detects free drug, drug in complex, or both, and whether its processing dissociates complexes.
- Confirm that concentrations use compatible units and represent the same compartment and sampling matrix.
- Check whether the model's reported prediction is \(C_L\), \(C_{LR}\), or \(C_L+C_{LR}\), rather than relying on an ambiguous variable name.
- Compare the time patterns of the measured quantity and the corresponding predicted quantity; a good fit to total drug does not establish that free drug is also well predicted.
A common mistake is to treat every reported plasma drug concentration as free drug and then interpret a slow decline as slow elimination of unbound drug. That conclusion can be wrong if the assay measures drug released from a bound pool. The correction is to establish the assay's drug-equivalent definition first, then compare it with the matching model quantity. A second mistake is to add free drug, complex concentration, and total drug together. Total drug already equals free plus complex-bound drug, so adding all three double-counts the drug.
Choosing the relevant concentration
The appropriate concentration depends on the question and the observation process. Free concentration is the direct choice when the quantity of interest is unbound drug in plasma. Total concentration is appropriate when the assay recovers both free and complex-bound drug, or when the intended summary is explicitly total drug equivalents. Complex concentration is necessary when the observation specifically quantifies drug–target complex. These choices are not competing definitions of one number; they represent distinct quantities.
Total concentration can be convenient for measurement, but it may conceal how drug is partitioned between free and bound forms. Free concentration preserves that distinction but may be harder to measure reliably. A mechanistic model can predict all three quantities, but predictions should be compared to data only after the observation process has been specified. Neither the total nor the free profile alone reveals the binding and removal mechanisms uniquely.
The bookkeeping identity also has clear limits. It assumes the reported total is measured in one compartment and counts drug molecules in the included forms. If other drug-containing species are present and detected, their drug-equivalent concentrations must also be included. For complexes with a stoichiometry other than one drug molecule per complex, the bound contribution must be multiplied by the number of drug molecules per complex. Finally, plasma total concentration should not be interpreted as total amount in the body without a model that accounts for distribution volumes and other compartments.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A hypothetical sample contains \(0.80\ \mathrm{nM}\) free drug and \(2.20\ \mathrm{nM}\) drug in 1:1 complex. What is total drug concentration in drug equivalents?
- Why do association and dissociation cancel when deriving the balance for total drug?
- A total-drug assay dissociates complexes during processing. Which model quantity should its result be compared with, assuming no other drug-containing forms are detected?
- In the rapid-equilibrium illustration, what happens to the bound pool as free concentration becomes much larger than \(K_D\)?
- Why is a plasma total concentration not automatically the total amount of drug in the body?
Solutions. (1) \(0.80+2.20=3.00\ \mathrm{nM}\). (2) Each association event removes one drug molecule from the free state and adds one to the complex state; dissociation does the reverse, so the terms have equal magnitude and opposite signs in the sum. (3) Compare with \(C_L+C_{LR}\), the total drug-equivalent concentration. (4) The bound pool approaches the available target concentration \(C_{R,\mathrm{tot}}\), under the stated equilibrium assumptions. (5) A concentration describes drug per volume in the sampled compartment; total body amount requires amounts across relevant compartments and their volumes.